The Grange School
Maths Department
Further Pure 2
OCR Past Papers
2
Jan 2006
1
(i) Write down and simplify the first three non-zero terms of the Maclaurin series for ln(1 + 3x).
[3]
(ii) Hence find the first three non-zero terms of the Maclaurin series for
ex ln(1 + 3x),
simplifying the coefficients.
[3]
2
Use the Newton-Raphson method to find the root of the equation e−x = x which is close to x = 0.5.
Give the root correct to 3 decimal places.
[5]
3
Express
4
Answer the whole of this question on the insert provided.
x+6
in partial fractions.
x(x2 + 2)
[5]
The sketch shows the curve with equation y = F(x) and the line y = x. The equation x = F(x) has roots
x = α and x = β as shown.
(i) Use the copy of the sketch on the insert to show how an iteration of the form xn+1 = F(xn ), with
starting value x1 such that 0 < x1 < α as shown, converges to the root x = α .
[3]
(ii) State what happens in the iteration in the following two cases.
(a) x1 is chosen such that α < x1 < β .
(b) x1 is chosen such that x1 > β .
[3]
4726/Jan06
Jan 2006
4
2
(i)
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(ii) (a)
...............................................................................................................................
.................
(b)
...............................................................................................................................
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Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be
pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations
Syndicate (UCLES), which is itself a department of the University of Cambridge.
4726/Ins/Jan06
3
Jan 2006
5
(i) Find the equations of the asymptotes of the curve with equation
y=
x2 + 3x + 3
.
x+2
[3]
(ii) Show that y cannot take values between −3 and 1.
6
[5]
(i) It is given that, for non-negative integers n,
1
In = e−x xn dx.
0
Prove that, for n ≥ 1,
In = nIn−1 − e−1 .
[4]
(ii) Evaluate I3 , giving the answer in terms of e.
[4]
7
√
The diagram shows the curve with equation y = x. A set of N rectangles of unit width is drawn,
starting at x = 1 and ending at x = N + 1, where N is an integer (see diagram).
(i) By considering the areas of these rectangles, explain why
√
N +1 √
√
√
√
1 + 2 + 3 + ... + N < x dx.
[3]
1
(ii) By considering the areas of another set of rectangles, explain why
√
N√
√
√
√
1+ 2+ 3+ ... + N > x dx .
[3]
0
N
(iii) Hence find, in terms of N , limits between which
∑
√
r lies.
[3]
r=1
4726/Jan06
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4
Jan 2006
8
The equation of a curve, in polar coordinates, is
r = 1 + cos 2θ ,
9
for 0 ≤ θ < 2π .
(i) State the greatest value of r and the corresponding values of θ .
[2]
(ii) Find the equations of the tangents at the pole.
[2]
(iii) Find the exact area enclosed by the curve and the lines θ = 0 and θ = 12 π .
[5]
(iv) Find, in simplified form, the cartesian equation of the curve.
[4]
(i) Using the definitions of cosh x and sinh x in terms of ex and e−x , prove that
sinh 2x = 2 sinh x cosh x.
[4]
(ii) Show that the curve with equation
y = cosh 2x − 6 sinh x
has just one stationary point, and find its x-coordinate in logarithmic form. Determine the nature
of the stationary point.
[8]
4726/Jan06
2
June 2006
1
Find the first three non-zero terms of the Maclaurin series for
(1 + x) sin x,
simplifying the coefficients.
2
[3]
1
dy
=
.
dx 1 + x2
(i) Given that y = tan−1 x, prove that
[3]
(ii) Verify that y = tan−1 x satisfies the equation
(1 + x2 )
3
4
The equation of a curve is y =
d2 y
dy
+ 2x
= 0.
2
dx
dx
[3]
x+1
.
x2 + 3
(i) State the equation of the asymptote of the curve.
[1]
(ii) Show that − 16 ≤ y ≤ 12 .
[5]
(i) Using the definition of cosh x in terms of ex and e−x , prove that
cosh 2x = 2 cosh2 x − 1.
[3]
(ii) Hence solve the equation
cosh 2x − 7 cosh x = 3,
giving your answer in logarithmic form.
5
[4]
(i) Express t2 + t + 1 in the form (t + a)2 + b.
[1]
(ii) By using the substitution tan 12 x = t, show that
1
π
2
0
√
1
3
dx =
π.
2 + sin x
9
4726/S06
[6]
June 2006
3
6
The diagram shows the curve with equation y = 3x for 0 ≤ x ≤ 1. The area A under the curve between
these limits is divided into n strips, each of width h where nh = 1.
(i) By using the set of rectangles indicated on the diagram, show that A >
(ii) By considering another set of rectangles, show that A <
2h
.
−1
[3]
3h
(2h)3h
.
3h − 1
[3]
(iii) Given that h = 0.001, use these inequalities to find values between which A lies.
7
8
[2]
The equation of a curve, in polar coordinates, is
√
r = 3 + tan θ , for − 13 π ≤ θ ≤ 14 π .
(i) Find the equation of the tangent at the pole.
[2]
(ii) State the greatest value of r and the corresponding value of θ .
[2]
(iii) Sketch the curve.
[2]
(iv) Find the exact area of the region enclosed by the curve and the lines θ = 0 and θ = 14 π .
[5]
The curve with equation y =
sinh x
, for x > 0, has one turning point.
x2
(i) Show that the x-coordinate of the turning point satisfies the equation x − 2 tanh x = 0.
[3]
(ii) Use the Newton-Raphson method, with a first approximation x1 = 2, to find the next two
[5]
approximations, x2 and x3 , to the positive root of x − 2 tanh x = 0.
(iii) By considering the approximate errors in x1 and x2 , estimate the error in x3 . (You are not expected
to evaluate x4 .)
[3]
[Question 9 is printed overleaf.]
4726/S06
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4
June 2006
9
(i) Given that y = sinh−1 x, prove that y = lnx + x2 + 1 .
[3]
(ii) It is given that, for non-negative integers n,
α
In = sinhn θ dθ ,
0
where α = sinh−1 1. Show that
√
nIn = 2 − (n − 1)In−2 ,
for n ≥ 2.
√
(iii) Evaluate I4 , giving your answer in terms of 2 and logarithms.
4726/S06
[6]
[4]
2
Jan 2007
1
It is given that f(x) = ln(3 + x).
(i) Find the exact values of f(0) and f (0), and show that f (0) = − 19 .
[3]
(ii) Hence write down the first three terms of the Maclaurin series for f(x), given that −3 < x ≤ 3.
[2]
2
It is given that f(x) = x2 − tan−1 x.
(i) Show by calculation that the equation f(x) = 0 has a root in the interval 0.8 < x < 0.9.
[2]
(ii) Use the Newton-Raphson method, with a first approximation 0.8, to find the next approximation
to this root. Give your answer correct to 3 decimal places.
[4]
3
2
The diagram shows the curve with equation y = ex , for 0 ≤ x ≤ 1. The region under the curve between
these limits is divided into four strips of equal width. The area of this region under the curve is A.
(i) By considering the set of rectangles indicated in the diagram, show that an upper bound for A
is 1.71.
[3]
4
(ii) By considering an appropriate set of four rectangles, find a lower bound for A.
[3]
(i) On separate diagrams, sketch the graphs of y = sinh x and y = cosech x.
[3]
(ii) Show that cosech x =
© OCR 2007
2ex
, and hence, using the substitution u = ex , find cosech x dx.
2
x
e −1
4726/01 Jan07
[6]
3
Jan 2007
5
It is given that, for non-negative integers n,
In = 1
π
2
xn cos x dx.
0
(i) Prove that, for n ≥ 2,
n
In = 12 π − n(n − 1)In−2 .
[5]
(ii) Find I4 in terms of π .
[4]
6
The diagram shows the curve with equation y =
2x2 − 3ax
, where a is a positive constant.
x2 − a 2
(i) Find the equations of the asymptotes of the curve.
[3]
(ii) Sketch the curve with equation
y2 =
2x2 − 3ax
.
x2 − a 2
State the coordinates of any points where the curve crosses the axes, and give the equations of
any asymptotes.
[5]
7
1 − t2
(i) Express 2
in partial fractions.
t (1 + t2 )
[4]
(ii) Use the substitution t = tan 12 x to show that
1
π
2
1
π
3
© OCR 2007
√
cos x
dx = 3 − 1 − 16 π .
1 − cos x
4726/01 Jan07
[5]
[Turn over
Jan 2007
8
(i) Define tanh y in terms of ey and e−y .
4
[1]
(ii) Given that y = tanh−1 x, where −1 < x < 1, prove that y = 12 ln
1+x
.
1−x
[3]
(iii) Find the exact solution of the equation 3 cosh x = 4 sinh x, giving the answer in terms of a
logarithm.
[2]
(iv) Solve the equation
tanh−1 x + ln(1 − x) = ln 45 .
9
[3]
The equation of a curve, in polar coordinates, is
r = sec θ + tan θ ,
for 0 ≤ θ ≤ 13 π .
(i) Sketch the curve.
[2]
(ii) Find the exact area of the region bounded by the curve and the lines θ = 0 and θ = 13 π .
[6]
(iii) Find a cartesian equation of the curve.
[3]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES),
which is itself a department of the University of Cambridge.
© OCR 2007
4726/01 Jan07
2
June 2007
1
The equation of a curve, in polar coordinates, is
r = 2 sin 3θ ,
for 0 ≤ θ ≤ 13 π .
Find the exact area of the region enclosed by the curve between θ = 0 and θ = 13 π .
2
(i) Given that f(x) = sin(2x + 14 π ), show that f(x) =
1
2
√
2(sin 2x + cos 2x).
[4]
[2]
(ii) Hence find the first four terms of the Maclaurin series for f(x). [You may use appropriate results
given in the List of Formulae.]
[3]
3
4
It is given that f(x) =
x2 + 9x
.
(x − 1)(x2 + 9)
(i) Express f(x) in partial fractions.
[4]
(ii) Hence find f(x) dx.
[2]
(i) Given that
find
y = x 1 − x2 − cos−1 x,
dy
in a simplified form.
dx
[4]
1 (ii) Hence, or otherwise, find the exact value of 2 1 − x2 dx.
[3]
0
5
It is given that, for non-negative integers n,
e
In = (ln x)n dx.
1
(i) Show that, for n ≥ 1,
In = e − nIn−1 .
(ii) Find I3 in terms of e.
© OCR 2007
[4]
[4]
4726/01 Jun07
3
June 2007
6
The diagram shows the curve with equation y =
unit width, starting at x = 1.
1
for x > 0, together with a set of n rectangles of
x2
(i) By considering the areas of these rectangles, explain why
1
1
1
1
+ 2 + 2 + ... + 2 > 2
1
2
3
n
n+1
1
1
dx.
x2
[2]
(ii) By considering the areas of another set of rectangles, explain why
1
1
1
1
+ 2 + 2 + ... + 2 < 2
2
3
4
n
n
1
1
dx.
x2
[3]
(iii) Hence show that
1
1−
<
n+1
∞
(iv) Hence give bounds between which
∑r
1
2
n
∑r
1
2
r=1
1
<2− .
n
lies.
[4]
[2]
r=1
7
(i) Using the definitions of hyperbolic functions in terms of exponentials, prove that
cosh x cosh y − sinh x sinh y = cosh(x − y).
(ii) Given that cosh x cosh y = 9 and sinh x sinh y = 8, show that x = y.
[4]
[2]
(iii) Hence find the values of x and y which satisfy the equations given in part (ii), giving the answers
in logarithmic form.
[4]
© OCR 2007
4726/01 Jun07
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June 2007
The iteration xn+1 =
8
x(x + 2)2 = 1.
4
1
, with x1 = 0.3, is to be used to find the real root, α , of the equation
(xn + 2)2
(i) Find the value of α , correct to 4 decimal places. You should show the result of each step of the
iteration process.
[4]
(ii) Given that f(x) =
9
1
, show that f (α ) ≠ 0.
(x + 2)2
[2]
(iii) The difference, δr , between successive approximations is given by δr = xr+1 − xr . Find δ3 .
[1]
(iv) Given that δr+1 ≈ f (α )δr , find an estimate for δ10 .
[3]
It is given that the equation of a curve is
y=
x2 − 2ax
,
x−a
where a is a positive constant.
(i) Find the equations of the asymptotes of the curve.
[4]
(ii) Show that y takes all real values.
[4]
(iii) Sketch the curve y =
x2 − 2ax
.
x−a
[3]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES),
which is itself a department of the University of Cambridge.
© OCR 2007
4726/01 Jun07
1
2
Jan 2008
It is given that f(x) = ln(1 + cos x).
(i) Find the exact values of f(0), f (0) and f (0).
[4]
(ii) Hence find the first two non-zero terms of the Maclaurin series for f(x).
[2]
2
y
1
2
P
O
x
1
The diagram shows parts of the curves with equations y = cos−1 x and y = 12 sin−1 x, and their point of
intersection P.
√
(i) Verify that the coordinates of P are 12 3, 16 π .
[2]
(ii) Find the gradient of each curve at P.
[3]
3
y
y = Ö1 + x 3
O
3
2
The diagram shows the curve with equation y =
between these limits has area A.
x
1 + x3 , for 2 ≤ x ≤ 3. The region under the curve
√
(i) Explain why 3 < A < 28.
[2]
(ii) The region is divided into 5 strips, each of width 0.2. By using suitable rectangles, find improved
lower and upper bounds between which A lies. Give your answers correct to 3 significant figures.
[4]
© OCR 2008
4726/01 Jan08
3
4
Jan 2008
The equation of a curve, in polar coordinates, is
r = 1 + 2 sec θ ,
for − 12 π < θ < 12 π .
(i) Find the exact area of the region bounded by the curve and the lines θ = 0 and θ = 16 π .
[5]
[The result sec θ dθ = ln | sec θ + tan θ | may be assumed.]
(ii) Show that a cartesian equation of the curve is (x − 2) x2 + y2 = x.
5
[3]
y
x
O
The diagram shows the curve with equation y = x e−x + 1. The curve crosses the x-axis at x = α .
(i) Use differentiation to show that the x-coordinate of the stationary point is 1.
[2]
α is to be found using the Newton-Raphson method, with f(x) = x e−x + 1.
(ii) Explain why this method will not converge to α if an initial approximation x1 is chosen such that
x1 > 1.
[2]
(iii) Use this method, with a first approximation x1 = 0, to find the next three approximations x2 , x3
and x4 . Find α , correct to 3 decimal places.
[5]
6
The equation of a curve is y =
2x2 − 11x − 6
.
x−1
(i) Find the equations of the asymptotes of the curve.
[3]
(ii) Show that y takes all real values.
[5]
© OCR 2008
4726/01 Jan08
[Turn over
4
Jan 2008
7
It is given that, for integers n ≥ 1,
1
In = 0
1
dx.
(1 + x2 )n
1
−n
(i) Use integration by parts to show that In = 2
+ 2n 0
8
x2
dx.
(1 + x2 )n+1
[3]
(ii) Show that 2nIn+1 = 2−n + (2n − 1)In .
[3]
(iii) Find I2 in terms of π .
[3]
(i) By using the definition of sinh x in terms of ex and e−x , show that
sinh3 x = 14 sinh 3x − 34 sinh x.
[4]
(ii) Find the range of values of the constant k for which the equation
sinh 3x = k sinh x
has real solutions other than x = 0.
[3]
(iii) Given that k = 4, solve the equation in part (ii), giving the non-zero answers in logarithmic form.
[3]
9
(i) Prove that
d
1
.
(cosh−1 x) = dx
x2 − 1
[3]
1
dx.
(ii) Hence, or otherwise, find 4x2 − 1
(iii) By means of a suitable substitution, find [2]
4x2 − 1 dx.
[6]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be
pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES),
which is itself a department of the University of Cambridge.
© OCR 2008
4726/01 Jan08
2
1
June 2008
It is given that f(x) =
fractions.
2ax
, where a is a non-zero constant. Express f(x) in partial
(x − 2a)(x2 + a2 )
[5]
2
y
5
–2
O
3
4
x
The diagram shows the curve y = f(x). The curve has a maximum point at (0, 5) and crosses the
x-axis at (−2, 0 ), (3, 0) and (4, 0). Sketch the curve y2 = f(x), showing clearly the coordinates of any
turning points and of any points where this curve crosses the axes.
[5]
3
By using the substitution t = tan 12 x, find the exact value of
1
π
2
0
1
dx,
2 − cos x
giving the answer in terms of π .
4
[6]
(i) Sketch, on the same diagram, the curves with equations y = sech x and y = x2 .
[3]
(ii) By using the definition of sech x in terms of ex and e−x , show that the x-coordinates of the points
at which these curves meet are solutions of the equation
x2 =
(iii) The iteration
2ex
.
e2x + 1
xn+1 =
2e
[3]
xn
e2xn + 1
can be used to find the positive root of the equation in part (ii). With initial value x1 = 1, the
approximations x2 = 0.8050, x3 = 0.8633, x4 = 0.8463 and x5 = 0.8513 are obtained, correct to
4 decimal places. State with a reason whether, in this case, the iteration produces a ‘staircase’ or
a ‘cobweb’ diagram.
[2]
5
It is given that, for n ≥ 0,
In = 1
π
4
tann x dx.
0
(i) By considering In + In−2 , or otherwise, show that, for n ≥ 2,
(n − 1)(In + In−2 ) = 1.
(ii) Find I4 in terms of π .
© OCR 2008
[4]
[4]
4726/01 Jun08
6
3
June 2008
7
It is given that f(x) = 1 − 2 .
x
(i) Use the Newton-Raphson method, with a first approximation x1 = 2.5, to find the next
approximations x2 and x3 to a root of f(x) = 0. Give the answers correct to 6 decimal places. [3]
(ii) The root of f(x) = 0 for which x1 , x2 and x3 are approximations is denoted by α . Write down the
exact value of α .
[1]
(iii) The error en is defined by en = α − xn . Find e1 , e2 and e3 , giving your answers correct to 5 decimal
e32
places. Verify that e3 ≈ 2 .
[3]
e1
7
It is given that f(x) = tanh−1 (i) Show that f (x) = −
1−x
, for x > − 12 .
2+x
1
, and find f (x).
1 + 2x
[6]
(ii) Show that the first three terms of the Maclaurin series for f(x) can be written as ln a + bx + cx2 ,
[4]
for constants a, b and c to be found.
8
The equation of a curve, in polar coordinates, is
r = 1 − sin 2θ ,
for 0 ≤ θ < 2π .
(i)
q =a
q =0
O
The diagram shows the part of the curve for which 0 ≤ θ ≤ α , where θ = α is the equation of the
[2]
tangent to the curve at O. Find α in terms of π .
(ii) (a) If f(θ ) = 1 − sin 2θ , show that f 12 (2k + 1)π − θ = f(θ ) for all θ , where k is an integer. [3]
(b) Hence state the equations of the lines of symmetry of the curve
r = 1 − sin 2θ ,
for 0 ≤ θ < 2π .
[2]
(iii) Sketch the curve with equation
r = 1 − sin 2θ ,
for 0 ≤ θ < 2π .
State the maximum value of r and the corresponding values of θ .
© OCR 2008
4726/01 Jun08
[4]
[Turn over
4
June 2008
9
(i) Prove that N
ln(1 + x) dx = (N + 1) ln(N + 1) − N , where N is a positive constant.
[4]
0
(ii)
y
O
68 69 70
1 2 3
x
The diagram shows the curve y = ln(1 + x), for 0 ≤ x ≤ 70, together with a set of rectangles of
unit width.
(a) By considering the areas of these rectangles, explain why
ln 2 + ln 3 + ln 4 + . . . + ln 70 < 70
ln(1 + x) dx.
[2]
0
(b) By considering the areas of another set of rectangles, show that
ln 2 + ln 3 + ln 4 + . . . + ln 70 > 69
ln(1 + x) dx.
[3]
0
(c) Hence find bounds between which ln(70!) lies. Give the answers correct to 1 decimal place.
[3]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be
pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES),
which is itself a department of the University of Cambridge.
© OCR 2008
4726/01 Jun08
2
Jan 2009
1
(i) Write down and simplify the first three terms of the Maclaurin series for e2x .
[2]
(ii) Hence show that the Maclaurin series for
ln(e2x + e−2x )
begins ln a + bx2 , where a and b are constants to be found.
2
[4]
It is given that α is the only real root of the equation x5 + 2x − 28 = 0 and that 1.8 < α < 2.
p
(i) The iteration xn+1 = 5 28 − 2xn , with x1 = 1.9, is to be used to find α . Find the values of x2 , x3
and x4 , giving the answers correct to 7 decimal places.
[3]
(ii) The error en is defined by en = α − xn . Given that α = 1.891 574 9, correct to 7 decimal places,
e
e
evaluate 3 and 4 . Comment on these values in relation to the gradient of the curve with
e2
e3
√5
[3]
equation y = 28 − 2x at x = α .
3
1
(i) Prove that the derivative of sin−1 x is p
.
1 − x2
(ii) Given that
find the exact value of
4
[3]
sin−1 2x + sin−1 y = 12 π ,
dy
when x = 14 .
dx
[4]
(i) By means of a suitable substitution, show that
ä p
can be transformed to ã cosh2 θ dθ .
(ii) Hence show that ä p
© OCR 2009
x2
x2 − 1
x2
x2 − 1
dx
[2]
p
dx = 12 x x2 − 1 + 12 cosh−1 x + c.
4726 Jan09
[4]
3
Jan 2009
5
y
2
a
O
1
b
g
x
The diagram shows the curve with equation y = f(x), where
f(x) = 2x3 − 9x2 + 12x − 4.36.
The curve has turning points at x = 1 and x = 2 and crosses the x-axis at x = α , x = β and x = γ , where
0 < α < β < γ.
(i) The Newton-Raphson method is to be used to find the roots of the equation f(x) = 0, with x1 = k.
(a) To which root, if any, would successive approximations converge in each of the cases k < 0
[2]
and k = 1?
(b) What happens if 1 < k < 2?
[2]
(ii) Sketch the curve with equation y2 = f(x). State the coordinates of the points where the curve
crosses the x-axis and the coordinates of any turning points.
[4]
6
(i) Using the definitions of cosh x and sinh x in terms of ex and e−x , show that
1 + 2 sinh2 x ≡ cosh 2x.
(ii) Solve the equation
cosh 2x − 5 sinh x = 4,
giving your answers in logarithmic form.
© OCR 2009
4726 Jan09
[3]
[5]
Turn over
4
Jan 2009
7
P (r, a)
Q
q=0
O
R
S
The diagram shows the curve with equation, in polar coordinates,
r = 3 + 2 cos θ ,
for 0 ≤ θ < 2π .
The points P, Q, R and S on the curve are such that the straight lines POR and QOS are perpendicular,
where O is the pole. The point P has polar coordinates (r, α ).
(i) Show that OP + OQ + OR + OS = k, where k is a constant to be found.
[3]
(ii) Given that α = 14 π , find the exact area bounded by the curve and the lines OP and OQ (shaded in
the diagram).
[5]
© OCR 2009
4726 Jan09
5
Jan 2009
8
y
O
x
1
2
3
n–1
n
The diagram shows the curve with equation y =
1
. A set of n rectangles of unit width is drawn,
x+1
starting at x = 0 and ending at x = n, where n is an integer.
(i) By considering the areas of these rectangles, explain why
1
1 1
+ + ... +
< ln(n + 1).
2 3
n+1
[5]
(ii) By considering the areas of another set of rectangles, show that
1+
1 1
1
+ + . . . + > ln(n + 1).
2 3
n
(iii) Hence show that
1
ln(n + 1) +
<
n+1
(iv) State, with a reason, whether
9
∞
∑ r < ln(n + 1) + 1.
1
r=1
1
∑ r is convergent.
r=1
A curve has equation
y=
where a is a positive constant.
n+1
[2]
[2]
[2]
4x − 3a
,
2(x2 + a2 )
(i) Explain why the curve has no asymptotes parallel to the y-axis.
[2]
(ii) Find, in terms of a, the set of values of y for which there are no points on the curve.
[5]
(iii) Find the exact value of ä
© OCR 2009
2a
a
4x − 3a
dx, showing that it is independent of a.
2(x2 + a2 )
4726 Jan09
[5]
2
June 2009
1
y
0.3
0.9
0.6
1.2
1.5
O
x
The diagram shows the curve with equation y = ln(cos x), for 0 ≤ x ≤ 1.5. The region bounded by
the curve, the x-axis and the line x = 1.5 has area A. The region is divided into five strips, each of
width 0.3.
(i) By considering the set of rectangles indicated in the diagram, find an upper bound for A. Give
the answer correct to 3 decimal places.
[2]
(ii) By considering another set of five suitable rectangles, find a lower bound for A. Give the answer
correct to 3 decimal places.
[2]
(iii) How could you reduce the difference between the upper and lower bounds for A?
2
3
Given that y =
x2 + x + 1
, prove that y ≥
(x − 1)2
1
4
for all x ≠ 1.
[4]
(i) Given that f(x) = esin x , find f ′ (0) and f ′′ (0).
[4]
(ii) Hence find the first three terms of the Maclaurin series for f(x).
4
5
Express
x3
in partial fractions.
(x − 2)(x2 + 4)
It is given that I = ä
1
π
2
0
[2]
[6]
cos θ
dθ .
1 + cos θ
(i) By using the substitution t = tan
1
1
θ,
2
show that I = ä 0
(ii) Hence find I in terms of π .
© OCR 2009
[1]
2
− 1 dt.
1 + t2
[5]
[2]
4726 Jun09
3
June 2009
6
Given that
ä
find the exact value of a.
7
1
0
p
1
16 + 9x
2
dx + ä
2
0
p
1
9 + 4x2
dx = ln a,
(i) Sketch the graph of y = coth x, and give the equations of any asymptotes.
[6]
[3]
(ii) It is given that f(x) = x tanh x − 2. Use the Newton-Raphson method, with a first approximation
x1 = 2, to find the next three approximations x2 , x3 and x4 to a root of f(x) = 0. Give the answers
correct to 4 decimal places.
[4]
(iii) If f(x) = 0, show that coth x = 12 x. Hence write down the roots of f(x) = 0, correct to 4 decimal
places.
[3]
8
(i) Using the definitions of sinh x and cosh x in terms of ex and e−x , show that
a2 + 1
(a) cosh(ln a) ≡
, where a > 0,
2a
[3]
(b) cosh x cosh y − sinh x sinh y ≡ cosh(x − y).
(ii) Use part (i)(b) to show that cosh2 x − sinh2 x ≡ 1.
[3]
[1]
(iii) Given that R > 0 and a > 1, find R and a such that
13 cosh x − 5 sinh x ≡ R cosh(x − ln a).
[5]
(iv) Hence write down the coordinates of the minimum point on the curve with equation
y = 13 cosh x − 5 sinh x.
[2]
9
(i) It is given that, for non-negative integers n,
Show that, for n ≥ 2,
In = ã
1
π
2
sinn θ dθ .
0
nIn = (n − 1)In−2 .
[4]
(ii) The equation of a curve, in polar coordinates, is
r = sin3 θ ,
© OCR 2009
for 0 ≤ θ ≤ π .
(a) Find the equations of the tangents at the pole and sketch the curve.
[4]
(b) Find the exact area of the region enclosed by the curve.
[6]
4726 Jun09